A001560 -id:A001560 - cp's OEIS frontend

A052001 Even partition numbers.

2, 22, 30, 42, 56, 176, 490, 792, 1002, 1958, 2436, 3010, 3718, 5604, 6842, 12310, 37338, 53174, 89134, 105558, 124754, 204226, 451276, 614154, 715220, 831820, 1300156, 1741630, 2012558, 2323520, 4087968, 7089500, 8118264, 12132164

Offset: 1

Patrick De Geest, Nov 15 1999

nonn

Intersection of A005843 and A000041; A059841(a(n)) * A167392(a(n)) = 1. [Reinhard Zumkeller, Nov 03 2009]

Reinhard Zumkeller, Table of n, a(n) for n = 1..1000

Cf. A000041, A001560, A005843, A052002, A052003, A059841, A167392, A213179.

a052001 n = a052001_list !! (n-1)
a052001_list = filter even a000041_list
-- Reinhard Zumkeller, Nov 03 2015

Select[PartitionsP[Range[100]], EvenQ] (* Jean-François Alcover, Mar 01 2019 *)

for(n=1, 100, if((k=numbpart(n))%2==0, print1(k", "))) \\ Altug Alkan, Nov 02 2015

a(n) = 2*A213179(n). - Omar E. Pol, May 08 2013

Offset corrected by Reinhard Zumkeller, Nov 03 2015

A052002 Numbers with an odd number of partitions.

Original entry on oeis.org

0, 1, 3, 4, 5, 6, 7, 12, 13, 14, 16, 17, 18, 20, 23, 24, 29, 32, 33, 35, 36, 37, 38, 39, 41, 43, 44, 48, 49, 51, 52, 53, 54, 56, 60, 61, 63, 67, 68, 69, 71, 72, 73, 76, 77, 81, 82, 83, 85, 87, 88, 89, 90, 91, 92, 93, 95, 99, 102, 104, 105, 107, 111, 114, 115, 118, 119, 121

Offset: 1

Patrick De Geest, Nov 15 1999

A052003(n) = A000041(a(n+1)). - Reinhard Zumkeller, Nov 03 2015

Also, numbers having an odd number of partitions into distinct odd parts; that is, numbers m such that A000700(m) is odd. For example, 16 is in the list since 16 has 5 partitions into distinct odd parts, namely, 1 + 15, 3 + 13, 5 + 11, 7 + 9 and 1 + 3 + 5 + 7. See Formula section for a proof. - Peter Bala, Jan 22 2017

			From _Gus Wiseman_, Jan 13 2020: (Start)
The partitions of the initial terms are:
  (1)  (3)    (4)     (5)      (6)       (7)
       (21)   (22)    (32)     (33)      (43)
       (111)  (31)    (41)     (42)      (52)
              (211)   (221)    (51)      (61)
              (1111)  (311)    (222)     (322)
                      (2111)   (321)     (331)
                      (11111)  (411)     (421)
                               (2211)    (511)
                               (3111)    (2221)
                               (21111)   (3211)
                               (111111)  (4111)
                                         (22111)
                                         (31111)
                                         (211111)
                                         (1111111)
(End)

Clark Kimberling, Table of n, a(n) for n = 1..1000
O. Kolberg, Note on the parity of the partition function, Math. Scand. 7 1959 377-378. MR0117213 (22 #7995).

Cf. A000041, A000700, A001560, A052001, A052003.
The strict version is A001318, with complement A090864.
The version for prime instead of odd numbers is A046063.
The version for squarefree instead of odd numbers is A038630.
The version for set partitions appears to be A032766.
The version for factorizations is A331050.
The version for strict factorizations is A331230.

import Data.List (findIndices)
a052002 n = a052002_list !! (n-1)
a052002_list = findIndices odd a000041_list
-- Reinhard Zumkeller, Nov 03 2015

N:= 1000: # to get all terms <= N
V:= Vector(N+1):
V[1]:= 1:
for i from 1 to (N+1)/2  do
  V[2*i..N+1]:= V[2*i..N+1] + V[1..N-2*i+2] mod 2
od:
select(t -> V[t+1]=1, [$1..N]); # Robert Israel, Jan 22 2017

f[n_, k_] := Select[Range[250], Mod[PartitionsP[#], n] == k &]
Table[f[2, k], {k, 0, 1}] (* Clark Kimberling, Jan 05 2014 *)

for(n=0, 200, if(numbpart(n)%2==1, print1(n", "))) \\ Altug Alkan, Nov 02 2015

From Peter Bala, Jan 22 2016: (Start)

Sum_{n>=0} x^a(n) = (1 + x)*(1 + x^3)*(1 + x^5)*... taken modulo 2. Proof: Product_{n>=1} 1 + x^(2*n-1) = Product_{n>=1} (1 - x^(4*n-2))/(1 - x^(2*n-1)) = Product_{n>=1} (1 - x^(2*n))*(1 - x^(4*n-2))/( (1 - x^(2*n)) * (1 - x^(2*n-1)) ) = ( 1 + 2*Sum_{n>=1} (-1)^n*x^(2*n^2) )/(Product_{n>=1} (1 - x^n)) == 1/( Product_{n>=1} (1 - x^n) ) (mod 2). (End)

Offset corrected and b-file adjusted by Reinhard Zumkeller, Nov 03 2015

A163096 Odd numbers with an even number of partitions.

Original entry on oeis.org

9, 11, 15, 19, 21, 25, 27, 31, 45, 47, 55, 57, 59, 65, 75, 79, 97, 101, 103, 109, 113, 117, 125, 129, 131, 133, 135, 137, 141, 147, 149, 151, 153, 163, 167, 171, 175, 179, 187, 191, 197, 205, 207, 213, 217, 227, 231, 241, 243, 245, 247, 253, 255, 265, 267, 271

Offset: 1

Omar E. Pol, Aug 09 2009

nonn

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. A000041, A001560, A005408, A154796, A163097.

Select[Range[1,500,2],EvenQ[PartitionsP[#]]&] (* Vincenzo Librandi, Mar 19 2012 *)

More terms from Sean A. Irvine, Oct 26 2009

A243935 Numbers m such that 5 divides A000041(m).

Original entry on oeis.org

4, 7, 9, 14, 18, 19, 23, 24, 27, 29, 34, 38, 39, 44, 49, 54, 58, 59, 61, 64, 66, 68, 69, 71, 74, 79, 82, 84, 89, 94, 97, 99, 103, 104, 109, 114, 119, 120, 124, 127, 128, 129, 130, 134, 136, 139, 140, 142, 144, 149, 154, 159, 163, 164, 165, 169, 170, 173, 174

Offset: 1

Bruno Berselli, Jun 15 2014

Bruno Berselli, Table of n, a(n) for n = 1..1000

Numbers m such that k divides A000041(m), where k is prime: A001560 (k=2), A083214 (k=3), this sequence (k=5), A243936 (k=7), A027827 (k=11), A071750 (k=13). For k composite: A237278 (k=4), A035700 (k=12).

[n: n in [1..200] | IsZero(NumberOfPartitions(n) mod 5)];

Select[Range[200], Mod[PartitionsP[#], 5] == 0 &]

is(n)=numbpart(n)%5==0 \\ Charles R Greathouse IV, Apr 08 2015

# From Peter Luschny in A000041
@CachedFunction
def A000041(n):
    if n == 0: return 1
    S = 0; J = n-1; k = 2
    while 0 <= J:
        T = A000041(J)
        S = S+T if is_odd(k//2) else S-T
        J -= k if is_odd(k) else k//2
        k += 1
    return S
[n for n in (0..200) if mod(A000041(n),5) == 0]

A163997 Primes with an even number of partitions.

Original entry on oeis.org

2, 11, 19, 31, 47, 59, 79, 97, 101, 103, 109, 113, 131, 137, 149, 151, 163, 167, 179, 191, 197, 227, 241, 271, 307, 317, 337, 347, 353, 359, 379, 383, 397, 409, 419, 431, 439, 449, 487, 503, 509, 521, 523, 541, 557, 563, 569, 571, 577, 599, 607, 631, 641, 643

Offset: 1

Omar E. Pol, Aug 09 2009

nonn

			11 is in the sequence because the number of partitions of 11 is equal to 56 and 56 is an even number.

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A000040, A000041, A001560, A163096, A163998.

Select[Prime[Range[200]],EvenQ[PartitionsP[#]]&] (* Harvey P. Dale, Jul 01 2019 *)

More terms from Sean A. Irvine, Oct 18 2009

A280288 Numbers n such that number of partitions of n is even and number of partitions of n into distinct parts is odd.

Original entry on oeis.org

2, 15, 22, 26, 40, 57, 70, 100, 117, 126, 176, 187, 247, 260, 532, 551, 590, 651, 715, 782, 925, 950, 1001, 1027, 1080, 1107, 1162, 1276, 1365, 1457, 1520, 1552, 1650, 1751, 1820, 1926, 2072, 2185, 2262, 2301, 2380, 2420, 2501, 2667, 2752, 2926, 3015, 3060, 3151, 3290, 3432, 3577, 3725, 3927, 4082, 4187, 4240, 4401

Offset: 1

Ilya Gutkovskiy, Dec 31 2016

Intersection of A001318 and A001560.

Numbers n such that A000035(A000041(n)) = 0 and A000035(A000009(n)) = 1.

			15 is in the sequence because we have:
------------------------------------
number of partitions = 176 (is even)
------------------------------------
15 = 15
14 + 1 = 15
13 + 2 = 15
13 + 1 + 1 = 15
12 + 3 = 15
12 + 2 + 1 = 15
12 + 1 + 1 + 1 = 15
11 + 4 = 15
11 + 3 + 1 = 15
11 + 2 + 2 = 15
11 + 2 + 1 + 1 = 15
11 + 1 + 1 + 1 + 1 = 15
...
------------------------------------------------------
number of partitions into distinct parts = 27 (is odd)
------------------------------------------------------
15 = 15
14 + 1 = 15
13 + 2 = 15
12 + 3 = 15
12 + 2 + 1 = 15
11 + 4 = 15
11 + 3 + 1 = 15
10 + 5 = 15
10 + 4 + 1 = 15
10 + 3 + 2 = 15
...

Eric Weisstein's World of Mathematics, Partition Function P, Partition Function Q
Index entries for related partition-counting sequences

Cf. A000009, A000035, A000041, A001318, A001560, A280289, A280290, A280291.

Select[Range[4500], Mod[PartitionsP[#1], 2] == 0 && Mod[PartitionsQ[#1], 2] == 1 & ]

A280290 Numbers n such that number of partitions of n is even and number of partitions of n into distinct parts is even.

Original entry on oeis.org

8, 9, 10, 11, 19, 21, 25, 27, 28, 30, 31, 34, 42, 45, 46, 47, 50, 55, 58, 59, 62, 64, 65, 66, 74, 75, 78, 79, 80, 84, 86, 94, 96, 97, 98, 101, 103, 106, 108, 109, 110, 112, 113, 116, 120, 122, 124, 125, 128, 129, 130, 131, 133, 135, 136, 137, 141, 142, 147, 149, 151, 153, 154, 158, 160, 163, 167, 170, 171, 174, 175, 179, 180

Offset: 1

Ilya Gutkovskiy, Dec 31 2016

Intersection of A001560 and A090864.

Numbers n such that A000035(A000041(n)) = 0 and A000035(A000009(n)) = 0.

			8 is in the sequence because we have:
-----------------------------------
number of partitions = 22 (is even)
-----------------------------------
8 = 8
7 + 1 = 8
6 + 2 = 8
6 + 1 + 1 = 8
5 + 3 = 8
5 + 2 + 1 = 8
5 + 1 + 1 + 1 = 8
4 + 4 = 8
4 + 3 + 1 = 8
4 + 2 + 2 = 8
4 + 2 + 1 + 1 = 8
4 + 1 + 1 + 1 + 1 = 8
3 + 3 + 2 = 8
3 + 3 + 1 + 1 = 8
3 + 2 + 2 + 1 = 8
3 + 2 + 1 + 1 + 1 = 8
3 + 1 + 1 + 1 + 1 + 1 = 8
2 + 2 + 2 + 2 = 8
2 + 2 + 2 + 1 + 1 = 8
2 + 2 + 1 + 1 + 1 + 1 = 8
2 + 1 + 1 + 1 + 1 + 1 + 1 = 8
1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 = 8
-------------------------------------------------------
number of partitions into distinct parts = 6 (is even)
-------------------------------------------------------
8 = 8
7 + 1 = 8
6 + 2 = 8
5 + 3 = 8
5 + 2 + 1 = 8
4 + 3 + 1 = 8

Eric Weisstein's World of Mathematics, Partition Function P, Partition Function Q
Index entries for related partition-counting sequences

Cf. A000009, A000035, A000041, A001560, A090864, A280288, A280289, A280291.

Select[Range[180], Mod[PartitionsP[#1], 2] == Mod[PartitionsQ[#1], 2] == 0 & ]

A306956 Sum over all partitions of n into distinct parts of the LCM of the parts.

Original entry on oeis.org

1, 1, 2, 5, 7, 15, 21, 39, 58, 90, 142, 218, 325, 465, 695, 948, 1411, 1977, 2883, 3940, 5415, 7422, 10126, 14091, 18947, 25666, 34282, 45890, 60710, 82211, 108510, 142960, 185271, 240595, 315158, 409231, 531967, 688689, 880997, 1126451, 1447754, 1849743

Offset: 0

Alois P. Heinz, Mar 17 2019

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..200
Wikipedia, Partition (number theory)

Cf. A000009, A000041, A001560, A040051, A052002, A181844, A319301.

b:= proc(n, i, r) option remember; `if`(i*(i+1)/2 b(n$2, 1):
seq(a(n), n=0..44);

b[n_, i_, r_] := b[n, i, r] = If[i(i+1)/2 < n, 0, If[n == 0, r, b[n, i-1, r] + b[n-i, Min[i-1, n-i], LCM[i, r]]]];
a[n_] := b[n, n, 1];
Table[a[n], {n, 0, 44}] (* Jean-François Alcover, Mar 20 2019, translated from Maple *)

a(n) mod 2 = A040051(n).
a(n) is even <=> n in { A001560 }.
a(n) is odd  <=> n in { A052002 }.

A331231 Numbers k such that the number of factorizations of k into distinct factors > 1 is even.

Original entry on oeis.org

6, 8, 10, 14, 15, 16, 21, 22, 26, 27, 33, 34, 35, 38, 39, 46, 51, 55, 57, 58, 62, 64, 65, 69, 74, 77, 81, 82, 85, 86, 87, 91, 93, 94, 95, 96, 106, 111, 115, 118, 119, 120, 122, 123, 125, 129, 133, 134, 141, 142, 143, 144, 145, 146, 155, 158, 159, 160, 161, 166

Offset: 1

Gus Wiseman, Jan 12 2020

nonn

First differs from A319238 in having 300.

The version for integer partitions is A001560.
The version for strict integer partitions is A090864.
The version for set partitions appears to be A016789.
The non-strict version is A331051.
The version for primes (instead of evens) is A331201.
The odd version is A331230.
Factorizations are A001055 with image A045782 and complement A330976.
Strict factorizations are A045778 with image A045779 and complement A330975.
The least number with n strict factorizations is A330974(n).
Cf. A001318, A045780, A045783, A318286, A328966, A330973, A330991, A330997, A331022, A331023/A331024, A331050, A331200, A331232.

strfacs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[strfacs[n/d],Min@@#>d&]],{d,Rest[Divisors[n]]}]];
Select[Range[100],EvenQ[Length[strfacs[#]]]&]

A243936 Numbers m such that 7 divides A000041(m).

Original entry on oeis.org

5, 10, 11, 12, 16, 18, 19, 24, 26, 27, 33, 37, 39, 40, 41, 47, 48, 52, 53, 54, 55, 61, 68, 75, 76, 82, 83, 89, 96, 97, 103, 110, 111, 117, 124, 125, 131, 138, 140, 145, 147, 152, 159, 166, 170, 173, 177, 180, 187, 191, 194, 201, 208, 213, 215, 222, 225, 229, 232

Offset: 1

Bruno Berselli, Jun 15 2014

Bruno Berselli, Table of n, a(n) for n = 1..1000

Numbers m such that k divides A000041(m), where k is prime: A001560 (k=2), A083214 (k=3), A243935 (k=5), this sequence (k=7), A027827 (k=11), A071750 (k=13). For k composite: A237278 (k=4), A035700 (k=12).

[n: n in [1..250] | IsZero(NumberOfPartitions(n) mod 7)];

Select[Range[250], Mod[PartitionsP[#], 7] == 0 &]

is(n)=numbpart(n)%7==0 \\ Charles R Greathouse IV, Apr 08 2015

# From Peter Luschny in A000041
@CachedFunction
def A000041(n):
    if n == 0: return 1
    S = 0; J = n-1; k = 2
    while 0 <= J:
        T = A000041(J)
        S = S+T if is_odd(k//2) else S-T
        J -= k if is_odd(k) else k//2
        k += 1
    return S
[n for n in (0..250) if mod(A000041(n),7) == 0]

cp's OEIS Frontend

A052001 Even partition numbers.

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A052002 Numbers with an odd number of partitions.

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A163096 Odd numbers with an even number of partitions.

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A243935 Numbers m such that 5 divides A000041(m).

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A163997 Primes with an even number of partitions.

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A280288 Numbers n such that number of partitions of n is even and number of partitions of n into distinct parts is odd.

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A280290 Numbers n such that number of partitions of n is even and number of partitions of n into distinct parts is even.

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Mathematica

A306956 Sum over all partitions of n into distinct parts of the LCM of the parts.